Veteran Education Transfer Plan Cover Sheet

Title of ETP Instruction through Architecture Name of IISME Fellow James C. Gaub Fellow’s year-round email gaub_cory@cupertino.k12.ca.us Sponsor Company NASA Ames Research Center Name of Mentor Denise Kiser National Board Certificate Area Career and Technical Education I, the IISME Fellow named above, affirm that the ETP I am submitting is my own work, that I acknowledged sources where appropriate, and that I avoided including any proprietary information of the Sponsor Company. By my submission I am assigning to IISME my entire copyright in the ETP. I understand IISME is simultaneously granting me a license to use the ETP for pedagogical purposes.

Signature Date Category: Data Based ETP Curriculum Subject: Math Science Technology _________ Level: Elem - Middle High Other Staff Development

Describe__________________________ Other Describe__________________________

Objectives Teaching objectives include increased teacher knowledge and ease of acquisition regarding effective science and mathematics instructional resources. Student objectives include further understanding of mathematics in classroom and global contexts, advancement of student performance in geometric applications, and facilitation of student interest in math and science career pursuits.

Abstract (50 words or less) The project will take the form of action research wherein the teacher will participate as researcher to inform instructional practices, and further student performance in science and mathematics. An established need exists for increased student achievement and understanding in science and math, specifically geometric applications within mathematics. Baseline data of current student performance in geometry will be measured with appropriate and specific metrics. After a results analysis is performed, a series of strategically designed instructional lessons will be implemented to improve student performance in targeted areas of need. A mid-point growth measure will be used to determine student progress performance. An additional metric of student performance will be applied to determine and measure student results after the instructional treatments are provided. Any needed remediation and/or further assessment in areas of poor performance will be administered. The final steps will include evaluations and reporting student results-analysis and research process to colleagues and IISME.

Describe how your ETP aligns with the National Board Standard stated in your proposal.

National Board for Professional Teaching Standards Career and Technical Education Standards Standard XI: Collaborative Partnerships Accomplished career and technical educators work with colleagues, the community, business and industry, and postsecondary institutions to extend and enrich the learning opportunities available to students and to ease school-to-work transitions. Over the next decade an entire generation of scientists and mathematicians will retire – a large gap in expertise will be left in the wake of this global trend. In order to dynamically advance knowledge, understanding, and strategic methods in science, math and technology learners in these fields will need to be trained and well equipped with greater efficacy than ever before. NASA has pioneered and championed exceptional advances in education, science, math, and technology only as NASA can over the past half-century and now stands poised to transport us forward into, and beyond the next. Recent customized curriculum development with NASA will provide applications for educators to promote interest and advances in mathematics and science through effective instructional design, and best practices that brings forward the highest results from teachers and students. The ETP aligns well with stated National Board Standard. The genesis and target-aim of the ETPwere developed through direct working partnerships with NASA educators, scientists, engineers, and management. The primary goals of the ETP are rooted in the enrichment of student learning opportunities, and these essential collaborative partnerships.

Describe the connection between your ETP and the Summer Fellowship.

As an IISME Fellow serving as a curriculum developer for NASA Ames Research Center I designed middle school mathematics curriculum to be used by the NASA Education Program. The curriculum is based on the NASA Personal Satellite Assistant (PSA), an autonomous spherical robot that will be used by astronauts onboard the International Space Station (ISS). The standards-based lessons and investigations will teach Geometric concepts in volume and surface area related to the packaging of the PSA. The lesson series is a component in a complete suite of lessons based on mathematic and scientific principles found in Geometry and Physics. A NASA Connect instructional film will be developed as part of the total project. Public Broadcasting Station will broadcast a television special about the autonomous PSA in a PBS learning series.

Growth-Measurement Devices

To measure student performance a collection of data sources will be acquired for analysis. Pre-performance measures will include California State Standardized Test results, and a standardized teacher-created baseline measurement in Geometry (Surface Area and Volume). Duplicate mid-point and end-point measurements in Geometry (Surface Area and Volume) will be utilized. An ongoing anecdotal records journal will be maintained throughout the process. The records journal will be coded for analysis as a final data source of student performance over time. Individual data sources and metrics will together present a complete diagnostic of student growth.

Resources Needed Materials required for implementation include standards-based mathematics lessons and curriculum supplies, California State Standardized Test results, standardized teacher-created baseline, mid-point, and end-point growth performance metrics, observational/anecdotal records journal.

Evaluation/Assessment Measures Used

California State standardized test scores, standardized teacher-created baseline, midpoint, and endpoint growth performance metrics, anecdotal records journal, mathematics inter-curriculum assessment measurements.

Formatting specifications

PC ____ or Mac__X__ (Must be in Word or Text Format) Software used ___NA______________

Submitted Copy Soft and hard copy due to peer coach by the end of the summer fellowship. Also, a copy of the cover sheet signed by a school site administrator submitted to IISME Oct.1, 2003 to receive $300 grant.

I, the Mentor named above [please select one of the following], q have read the attached ETP, and my comments, if any, appear below. q have read the attached ETP, and, as outlined in the IISME-Company Fellowship Agreement, have reviewed it

on behalf of the Sponsor Company, and have determined that the ETP does not contain any Sponsor-proprietary information. My additional comments, if any, appear below.

Comments:

Signature Date Administrator’s comments:

Signature Date

Instruction

through

Architecture

Training & Curriculum in Mathematic Applications for Strategic Support of the Personal Satellite Assistant & the International Space Station

Industry Initiatives for Science and Math Education

An IISME Educational Transfer Plan

Prepared by Cory Gaub

IISME Fellowship – NASA Ames Research Center NASA Educational Technology

2003 National Board for Professional Teaching Standards

Career and Technical Education Standards Standard XI: Collaborative Partnerships

Accomplished career and technical educators work with colleagues, the community, business and industry, and postsecondary institutions to extend and enrich the learning opportunities available to students and to ease school-to-work transitions.

Teaching Outcomes

Teaching outcomes include increased teacher knowledge and ease of acquisition regarding effective science and mathematics instructional resources. Student outcomes include further understanding of Mathematics in discrete and global contexts, advancement of student performance in Geometric applications, and facilitation of student interest in math and science career pursuits. Abstract

Rational

Research indicates support for the development of these materials which correlates with the evident need for further development of student understanding in science and mathematics, specifically geometric applications. A paramount and more expansive need exists as well, which is to facilitate overall student interest in mathematics and science to promote interest and pursuit in math and science careers. Over the next decade an entire generation of scientists and mathematicians will retire – a large gap in expertise will be left in the wake of this global trend. In order to dynamically advance knowledge, understanding, and strategic methods in science, math and technology learners in these fields will need to be trained and well equipped with greater efficacy than ever before. NASA has pioneered and championed exceptional advances in education, science, math, and technology only as NASA can over the past half-century and now stands poised to transport us forward into, and beyond the next. Recent customized curriculum development with NASA will provide applications for educators to promote advances in mathematics and science through effective instructional design, and best practices, that brings forward the highest results from teachers and students. Design The inquiry based mathematics curriculum is effectively designed for implementation and dynamic application. Crafted and customized specifically for the NASA

Education Program, the curriculum framework maintains a fluid adaptability for a variety of solution-needs applications. The project will take the form of action research wherein the teacher participates as researcher to further student performance in science and mathematics. An established need exists for increased student achievement and understanding in science and math, specifically geometric applications within mathematics. Baseline data of current student performance in geometry will be measured with an appropriate and specific metric. After a results analysis is performed a series of strategically designed instructional lessons will be applied to improve student performance in target areas of need. A mid-point growth measure will be implemented to determine student progress performance. An additional metric of student performance will be applied to determine and measure student results after the instructional treatments are provided. Any needed remediation and further assessment in areas of poor performance will be administered. The final steps will be to analyze and report student results-analysis, and research process and results to colleagues and IISME. Materials & Measurements Materials required for implementation include Mathematics Curriculum Suite developed for NASA Personal Satellite Assistant (PSA) Project; student California State Standardized Test results; teacher created baseline, mid-point, and end-point growth performance metrics; observational/anecdotal records journal. Assessment Metrics To measure student performance a collection of data sources will be acquired for analysis. Pre-performance measures will include California State Standardized Test results, and a standardized teacher-created baseline measurement in Geometry (Surface Area and Volume). Duplicate mid-point and end-point measurements in Geometry (Surface Area and Volume) will be utilized. An ongoing anecdotal records journal will be maintained throughout the process. The records journal will be coded for analysis as a final data source of student performance over time. Individual data sources and metrics will together present a complete diagnostic of student growth. Project base The curriculum suite is based on the NASA Personal Satellite Assistant (PSA), an autonomous spherical robot that will be used by astronauts onboard the International Space Station (ISS). The curricular lessons and investigations will teach geometric concepts in surface area and volume related to the safe design, assembly, and functionality of the robotic PSA. A NASA Connect instructional film will be developed as part of the curriculum suite that includes other lessons on micro-gravity physics and systems onboard the ISS. A final televised learning component will compete the educational project. PBS will broadcast a television special about the autonomous PSA as part of NASA Education and a Public Broadcasting Station learning series.

NASA Educational Technology Mathematics Curriculum

~ Content & Design ~ § Main Lesson Concept

§ Scientific Question

§ Instructional Objectives

§ National Standards & Benchmarks

§ Misconceptions

§ Prerequisite Concepts

§ Major Concepts

§ Suggested Timeline

§ Equipment & Materials

Engage………………………….………………………….anticipatory set Explore………………………………………...…………. inquiry, observation Explain…………………………..………………………...data justification Extend/Apply…………………………...………………….performance task Evaluate..…………………………………………………assessment application Lesson 1: Surface Area Nets & Orthographic Mapping Lesson 2: Volume of Rectangular Prisms and Cubes Lesson 3: Surface Area & Volume Formula Applications Lesson 4: Dimensional Change Effects on Surface Area and Volume Metrics: Evaluative Rubric Criteria Guide

Curriculum Framework & Road Map § Concepts shown are lesson-specific. Given that conceptual understandings from lesson

1 are carried as prior knowledge into lesson 2, etc. Lesson objectives and standards will vary per lesson.

Lesson 1: Surface Area Nets Concepts: Students will understand two-dimensional representations of three-

dimensional objects can be used to visualize and solve problems such as those involving surface area and volume.

Standards: (PREREQ) NCTM 3 3-5 #4 2061 9C 3-5 #1, #4, #5

(MEETS) NCTM 3 6-8 #4.2, #4.4, #4.5 2061 9C 6-8 #1

(ADDRESS) NCTM 3 9-12 #1.1, #1.2

Objectives: Students will make and label 2-dimensional geometric nets for rectangular

solids/cubes using cardboard boxes and drawing paper. Students will make and label 3-dimensional solids using nets and

visualization. Students will use the nets to determine surface area. Lesson 2: Volume of Rectangular Prisms and Cubes Concepts: Students will understand that description, classification, and understanding of

geometric models can be used to represent and explain numerical and algebraic relationships. Students will understand relationships among types of two- and three-dimensional objects using their defining properties.

Standards: (PREREQ) NCTM 3 3-5 #1, #4 2061 9C 3-5 #1, #5

(MEETS) NCTM 3 6-8 #1.1, #4.2, #4.4, #4.5; NCTM 8 6-8 #1 2061 9C 6-8 #1 (ADDRESS) NCTM 3 9-12 #1.1, #4.1 2061 9C 9-12 #1

Objectives: Students will find surface area of cereal boxes using rulers and determine

volume of cereal boxes using solid cubes. Students will communicate conceptual understanding of surface area & volume.

Lesson 3: Surface Area and Volume Formula Concepts: Students will understand formula development strategies to determine the

surface area and volume of prisms. Students will understand techniques and tools to accurately find length, area,

and volume. Standards: (PRERQ) NCTM 4 3-5 #2 2061 9C 3-5 #1, #5, #6

(MEETS) NCTM 4 6-8 #2.2, #2.4; NCTM 8 6-8 #1, #4 2061 9C 6-8 #1

(ADDRESS) NCTM 4 9-12 #1 2061 9C 9-12 #1, #2 Objectives: Students will apply surface area formula and volume formula for cubes and

rectangular solids using cardboard boxes as geometric models. Students will collaborate and precisely express their understanding with peers and teachers using the language of mathematics.

Lesson 4: Dimensional Change Effects on Surface Area and Volume Concepts: Students will learn when linear dimensions of a rectangular solid change the

volume and surface area change disproportionately. Standards: (PREREQ) NCTM 3 3-5 #4; NCTM 4 3-5 #1, #2 2061 9C 3-5 #1, #4, #5, #6

(MEETS) NCTM 3 6-8 #2.4, #4.3, #4.4; NCTM 8 6-8 #1 – 4 2061 9C 6-8 #1, #3, #6

(ADDRESS) NCTM 3 9-12 #4.1, #4.2

2061 9C 9-12 #1, #2, #3 Objectives: Students will arrange-to-fit specific 2-dimensional shapes within 12” and 8”

graphed circles. Students will construct, and arrange-to-fit 3-dimensional paper solids within

1-Liter and 20-ounce plastic bottles. Students will identify how surface area and volume changed after a size reduction. Students will accurately record and report mathematical applications and findings.

LESSON 1: NASA CONNECT PSA VIDEO CHALLENGE Main Lesson Concept Two-dimensional representations of three-dimensional objects can be used to visualize and solve problems such as those involving surface area and volume. Scientific Question How can orthographic maps (nets) be used to visualize and model properties of geometric figures? Instructional Objectives Students will make and label 2- and 3-dimensional geometric nets for rectangular solids using cardboard boxes and drawing paper. Students will use the nets to determine surface area. National Standards Prerequisite NCTM: 3 3-5 #4 2061: 9C 3-5 # 1, #4, #5 Meets NCTM: 3 6-8 #4.2, #4.4, #4.5 2061: 9C 6-8 #1 Addresses NCTM: 3 9-12 #1.1, #1.2 Misconception Volume & surface area change proportionally. Students often believe that if volume increases, surface area increases by the same amount. Conversely, that if surface area increases, volume increases by the same amount. Displaying two cubes can show this. Each has a volume and surface area; however when the two cubes are placed together the volume remains constant while the surface area has decreased. Prerequisite Concepts Area can be thought of as a collection of unit squares.

Major Concepts Some shapes have special properties. Shapes can match exactly or have the same shape in different sizes. Visualization, spatial reasoning, and geometric modeling can solve problems. Suggested Timeline One class session. Materials and Equipment Each student PSA Team member (teams of 3-4) will need pre-drawn nets of rectangular prisms, boxes (envelope boxes, cereal boxes, etc.) of various shapes and sizes for each team, Math Journal for drawing and recording data. Teacher will need chalk/white board, boxes for display and demonstration, scissors and tape, gift wrapped present/box, student survey. Engage Actually perform this task in front of the class. This real life application will develop the concept of nets and views and a working vocabulary of: front view, back view, side view, top view, bottom view, and net. Discuss Suppose we wanted to unwrap a covered package to see what was inside.

As we unwrap it we are going to unfold the package and lay it out flat. First we are going to carefully take the wrapping paper off trying not to tear it. Then we are going to set it aside and talk just about the package inside right

now. Question When you look at this part (show front view) of the package what shape do you see? Encourage participation. Answer Rectangle.

Question What part of the package is that shape? Answer Front Teacher Action Draw that size of a rectangle on the board and label it “Front View.” Continue this with each face of the package until all faces are drawn and labeled on the board.

Front View

Back View

Left Side

Right Side

Top View

Bottom View

Teacher Action Cut the box so that it is completely unfolded and still connected. Explain Notice how the package should be completely unfolded and still

connected. The unfolded package is an example of a net.

Teacher Action Distribute boxes, scissors, & tape. Instruction Cut your boxes apart into pieces like pictured on the board. Explain & Instruct Now once you have all the pieces of a package it’s like putting a puzzle together. When assembling the PSA NASA scientists and engineers had

the same challenge. Take the pieces and lay them on a table or desk edge-to-edge. As the pieces are put back together discuss the reason why a top view edge and a bottom view edge cannot be placed beside each other. This package

is a prism, a 3-Dimensional shape, there are sides that fit in between the top and bottom faces. Teacher Action/Instruction Draw a 3-Dimensional rectangular prism on the board and talk students through how to draw unseen faces. Draw the views and label them. Demonstrate how to put the pieces into a net without cutting them. Emphasize visualizing it rather than cutting and pasting.

Explain There is more than one correct way to draw a net for a specific prism. We are going to experiment with some of those ways now. In order to complete the PSA NASA scientists and engineers had to

consider varieties of solutions for the same problem. Explore

Guided Activity Teacher Action Distribute copies of pre-drawn rectangular prism nets.

Student Action Cut out, fold, and tape to form rectangular prism. Cut out, cut into pieces, re-form net. Teacher Action Distribute copies of cube nets. Student Action Cut out, fold, and tape to form the cube. Cut out, cut into pieces, re-form net. Question What do you notice about the relationship between the two-dimensional (flat) nets and the 3-dimenstional nets? Answer Each face of the 3-dimensional shape is one section of the net. The location of a given face on the 3-dimensional shape is dependent on the location of a given section of the net. Encourage discussion/questions. Teacher Action/Instruction Teacher distributes cereal boxes, envelope boxes, etc. Have student

teams draw and label the views they see (front, back, top, bottom, left side, right side), and draw a net for their particular box. Student Action Draw and label the views of the box. Draw a net for the particular box. Note to Teacher Students may need to cut out the box faces and tape them back together. Teacher Action/Instruction Have student teams trade boxes and repeat drawing process of box views and nets using a differently shaped rectangular prism. Explain

Question What similarities and differences do you observe between the cube and a rectangular prism? Answer Cubes - all faces are the same size, Rectangular Prisms - faces are

different sizes. Explain/Question NASA PSA scientists and engineers used models that represent the PSA. How might that be similar to what we’ve done? Answer We used boxes as geometric models to solve problems. Part of solving

the problem for NASA (and for us) involved putting a model together like a puzzle. Encourage discussion. Note to teacher More complex pre-drawn nets may be attempted depending on

level of class and time permitted. Teacher Action Draw another 3-D shape(s) on the board for students to observe. Student Action Students draw the views and net(s) of the shape(s) without cutting and pasting together. Teacher Action Elicit from student strategies they used for drawing views and nets without cutting and pasting.

Extend/Apply Constructed Response Performance Task

Teacher Action Display a cereal box and ask students to draw and label the views they

see. (Show front, back, top, bottom, left side, and right side). Student Action Draw the net of the box directly underneath the views they have just drawn. If they cannot visualize it, they may need to cut the pieces out and tape

them back together. Teacher Action Display differently shaped boxes. Instruct students to draw and label the

net for each individual box. Note to Teacher Emphasize visualizing over cutting and pasting per NCTM

Standards. Teacher Action Encourage student discussion regarding visualization strategies for

drawing nets. Evaluate Teacher Action Distribute student inventory. Student Action Demonstrate your understanding of nets and views by drawing the front

view, the side views, the top view, and the bottom view of the figure. Label them with the words front, top, bottom, and side. Also label the dimensions of each piece. If possible, draw these connected as a net.

Explain how the idea of nets views would be important to NASA scientists and engineers when they developed and built the PSA. Explain how the

idea of visualization and geometric modeling would be important to them. Teacher Action/Assessment Anecdotal observation of student understanding. Review and provide assessment of student inventories & Math Journals using rubric criteria. Addendum

Student Inventory

1. State how to determine the area of a rectangle. ____________________ 2. State in your own words what surface area is and what it means to you. 3. Find the area of a rectangle with the following dimensions: 5 cm x 7 cm. 4. Explain what the following mathematical terms mean. Nets- Base Area- Side Area-

Front View- Side View- Back View-

Demonstrate your understanding of views by drawing the following: front view, side views, the top view, and the bottom view of the figure shown. Label them with the words front, top, bottom, and side. Label the dimensions of each piece. Draw these connected as a net.

Explain how the idea of nets and views would be important to NASA scientists and engineers when they built and developed the PSA. Explain how the idea of visualization and geometric modeling would be important to them. LESSON 2: NASA CONNECT PSA VIDEO CHALLENGE Main Lesson Concept Description, classification, and understanding of geometric models can be used to represent and explain numerical and algebraic relationships. Scientific Question How can problems in surface area and volume be solved with geometric models? Instructional Objectives Students will determine surface area and volume of rectangular solids using measurement and manipulatives. Students will communicate conceptual understanding of surface area and volume.

National Standards Prerequisite NCTM: 3 3-5 #1, #4 2061: 9C 3-5 # 1, #5 Meets NCTM: 3 6-8 #1.1, #4.2, #4.4, #4.5; NCTM 8 6-8 #1 2061: 9C 6-8 #1 Addresses NCTM: 3 9-12 #1.1, #4.1 2061: 9C 9-12 #1 Misconception Volume & surface area change proportionally. Students often believe that if volume increases, surface area increases by the same amount. Conversely, that if surface area increases, volume increases by the same amount. Displaying two cubes can show this. Each has a volume and surface area; however when the two cubes are placed together the volume remains constant while the surface area has decreased. Prerequisite Concepts Area can be thought of as a collection of unit squares. Volume can be thought of as a set of unit cubes. Formulas can be used to determine perimeter and area. Major Concepts Relationships among types of two-and three-dimensional objects can be determined and expressed using their defining properties. Suggested Timeline One class session. Materials and Equipment Each student PSA Team (teams of 3-4) will need: Centimeter ruler, 2 (unopened) single-serving cereal boxes, 30 one-centimeter manipulative cubes, scissors, centimeter graph paper, Math Journal for recording estimates. Teacher will need chalk/white board, single serving boxes for display and demonstration, scissors, and 300 one-centimeter cubes (sugar cubes can substitute).

Engage Discuss NASA scientists and engineers are pushing technology like never before with a new robot helper being used on the International Space Station

called the Personal Satellite Assistant or PSA. The PSA has amazing computer technology cased inside its interestingly protective package. We can think about the PSA and NASA technology to help us understand the ideas of geometric surface area and volume. Review The cereal, and other boxes we used in the previous lesson packaged and protected the various items that were inside them. Similarly, the PSA is not only a useful helper but also a package that holds and safely protects its contents.

The sum of the area of all the views, or faces on the box is known as thesurface area. It’s labeled just as the area of a rectangle or square would be (units squared or square units). The amount of space that is inside the box is known as the volume. It can be determined by finding out how many three dimensional cubes will fit inside.

Question How might surface area and volume be important to NASA scientists and engineers when they developed and built the PSA? Encourage student discussion and elicit student responses. Answer NASA would need to be able to determine how much space was available for the contents of the PSA and how to package it most effectively. NASA scientists and engineers would want to know how big or small the PSA was going to be. NASA would need to know how much they could spend on supplies and materials. Teacher Action Distribute single serving cereal boxes.

Question Why are packages the size and shape they are? Encourage student discussion and elicit student responses. Answer Various contents have different needs for protection and storage. Some shapes have special properties. Shapes can match exactly or have the same shape in different sizes. Explain NASA scientists and engineers had to consider these ideas when

designing and building the PSA. Explore

Guided Activity Teacher Action Distribute scissors, centimeter graph paper, centimeter ruler. Student Action

Cut out cereal boxes so they form one template (connected pattern) for the box when flattened out. Question How is the two-dimensional flat shape you have now like a net? Elicit student responses. Answer The shape is a net without labeled views. It can be folded up into a rectangular prism. Instruct Trace the pattern onto the centimeter graph paper and label each view, or face on the graph paper to show front, back, left, right, top, and bottom sides.

Student Action Place cut out box onto graph paper, trace, and label each face on the

graph paper with front, back, left, right, top and bottom sides. Instruct To the nearest whole centimeter make and record dimension (length,

width) estimates of each face and share your estimates within your team. Allow

for few minutes of team estimation & discussion. Teacher Action Elicit student estimates from each group. Instruction Have students measure and record the length and width of each face to

the nearest whole centimeter on the centimeter graph paper. Student Action Use a centimeter ruler to measure and record the length and width of

each face on the graph paper. Question What can we determine if we know the dimension measurements of

length and width. Answer Area and perimeter. Instruction Have students find the area of each face by multiplying the length times

the

width. (Students may need to be reminded that (Area = length x width) Have students sum and record in Math Journals the areas of all six faces to determine the total surface area. Explain

Area is measured in square units, or units squared because it shows the measurement of two-dimensional, or flat surface dimensions of length and width. Student Action Multiply the recorded length and width of each face to determine the area

of each face. Sum the areas of all six faces to determine the total surface

area of the rectangular cereal box. Explain Surface area is measured in square units, or units squared because it

shows the sum total measurements of two-dimensional, or flat surface

dimensions of length and width that covers an object. Surface area is very useful.

Here are a couple of examples: Examples If had to paint my room I’d want to know how much paint I’d need to

cover the walls – if I knew the surface area of the room I would know how

much paint to buy. If I was going to wrap a birthday gift box in special paper I’d want to know how much paper I’d need to cover the outsides of the box – knowing the surface area (sum total of all the areas of all of the sides) would tell me how much wrapping paper I’d need.

Teacher Action Have students discuss the following question within their team. Have students report out their responses and justify them with evidence. Allow several minutes for discussion & reporting. Question Why would NASA scientists and engineers want to understand the idea of surface area regarding the design and assembly of the PSA? Answer NASA scientists and engineers need to understand surface area to know

how much red paint to use, the amount of needed material for coverage, etc… Teacher Action Elicit student responses and encourage discussion. (Remember that the

big idea behind surface area is covering, wrapping, or packaging

something). Distribute a second set of single serving cereal boxes, and approximately 30 one-centimeter cubes for determining volume. Instruct & Demonstrate Cut open one of the large faces on the box along three of its edges and remove the inner bag of cereal. Estimate and record the number of one centimeter cubes that will fit inside the box. Fill the single serving cereal box with one-centimeter cubes. Student Action Cut open one of the large faces on the box along three edges and remove

the inner cereal bag. Estimate how many one-centimeter cubes will fit inside the box. Record your estimates.

Fill the single serving cereal box with one-centimeter cubes. Note to Teacher Some students may notice patterns in the layering of the cubes and

will not need to fill the whole box. If cubes are unavailable sugar cube

work well. Explain Question & Discussion What similarities and differences do you observe thus far between surface area and volume? Elicit student responses. Answer Surface area and volume show measurement of two- and three-

dimensional shapes respectively. Surface area is like covering something. Volume is

like holding something. Teacher Action Elicit student response and encourage discussion. Question How many cubes does the cereal box hold? Answer Most single-serving cereal boxes measure about 10 cm x 7 cm x 4 cm Question What does the number of cubes that the box can hold tell us? Answer The volume, or amount of “holding space” available in the cereal box. Explain

Volume is very useful. Here is an example: If I was going to send boxes of movies and video games to my

students I’d want to know how much volume or available space was in the boxes to

know how many movies and video games I could fit in each box. Question Why would NASA scientists and engineers want to understand the idea of volume regarding the design and assembly of the PSA (a spherical, ball-

like shape being packaged with cube or rectangular shapes)? Answer NASA scientists and engineers want to understand the idea of volume

since they had to determine how much space was available for supplies and materials, how large or small the PSA would be, etc… Teacher Action Elicit student responses and encourage discussion. (Remember that the

big idea behind volume is holding something). Extend/Apply Constructed Response Performance Task Explain We learned that when we find the area of each of the six faces (front,

back, left, right, top, bottom) on the rectangular prism (single serving cereal box), and then we sum the areas of all six faces of the rectangular prism (single serving cereal box) we can determine the total surface area. Question What is surface area? How do we measure it? Answer The area of all the faces of an object. It is measured by finding the area of

each face, then summing the areas of all six faces. Teacher Action For the following questions elicit student responses and encourage discussion. Student Action Students record solutions in Math Journals. Note to Teacher Questions facilitate understanding for determining volume formula. Question How many cubes are in each layer within the box? How many cubes are in each row? How many columns of cubes fit inside the box?

Explain The number of cubes in each layer tells us height (H) in centimeters. The number of cubes in each row tells us width (W) in centimeters.

The number of columns of cubes tells us length (L) in centimeters. These are the measurements we use to determine volume. When we put those measurements into the formula V = L x W x H we are able to know the volume of our single-serving cereal box. Teacher Action Encourage students to apply this formula using the information they just recorded and discuss within their teams. Student Action Discuss results of the volume of their single serving cereal boxes. Evaluate Explain In your math journals use pictures, numbers, and words to describe and explain: The concepts of surface area and volume. How surface area and volume can be determined. How and where surface area and volume are be useful in daily life.

How understanding surface area and volume is important to NASA scientists and engineers who designed and built the PSA. Student Action Respond in Math Journals then discuss responses within their teams. Allow time for students to write, discuss and elaborate ideas regarding surface area, volume, and NASA PSA applications. Teacher Action/Assessment Teacher anecdotal observation of student discussions. Review and

provide assessment of Math Journals using rubric criteria. Student Action Each Team collaborates and prepares a brief summary statement that evidences their mathematical understanding of surface area, volume and

it’s implications for NASA PSA. Teacher Action/Assessment Teacher anecdotal observation and rubric evaluation of team summary statements.

LESSON 3: NASA CONNECT PSA VIDEO CHALLENGE Main Lesson Concept Precise formulas, strategies, and techniques can be used with geometric tools to accurately determine surface area and volume of prisms. Scientific Question How can mathematical formulas be used to understand the relationship between volume and surface area? Instructional Objectives Students will determine the surface area and volume of rectangular prisms using mathematical formulas. Students will collaborate and definitively express their understanding with peers and teachers using the language of mathematics. National Standards Prerequisite NCTM: 4 3-5 #2 2061: 9C 3-5 #1, #5, #6 Meets NCTM: 4 6-8 #2.2, #2.4; NCTM: 8 6-8 #1, #4 2061: 9C 6-8 #1 Addresses NCTM: 4 9-12 #1 2061: 9C 9-12 #1, #2 Misconception Volume & surface area change proportionally. Students often believe that if volume increases, surface area increases by the same amount. Conversely, that if surface area increases, volume increases by the same amount. Displaying two cubes can show this. Each has a volume and surface area; however when the two cubes are placed together the volume remains constant while the surface area has decreased.

Prerequisite Concepts Area can be thought of as a collection of unit squares. Volume can be thought of as a set of unit cubes. Formulas can be used to determine perimeter and area. Areas of irregular shapes can be found by dividing them into squares and triangles. Major Concepts Some shapes have special properties. Shapes can match exactly or have the same shape in different sizes. Organized mathematical language is used to clearly communicate precise mathematical ideas to others. Suggested Timeline One class session. Materials and Equipment Students teams of 3-4 will need 2-3 small standardized rectangular boxes (single serve cereal boxes) scissors, tape, drawing paper, graph paper,

metric rulers, and Math Journals. Teacher will need observational records journal. Engage Discuss Within your lifetime it is highly probable that people will travel to Mars and possibly even establish human supported stations there! So imagine yourself in your new career as a pioneering NASA scientists or engineer who will help make this happen. As we’ve seen math and

science understanding will be important to help achieve this exciting opportunity for exploration. Essential understanding of volume and surface area are important for scientists and engineers working on the PSA for the ISS - the ideas we’ve been looking at with surface area and volume will also be essential for us. Review prior lesson 2 Thinking back to our recent investigations let’s talk about the surface area

and volume of a rectangular prism. Instruction Explain to your team what we discovered about the surface area of rectangular prisms in the prior investigation. Note to Teacher Allow time for students to discuss, monitor dialogue - some students may need instructional prompting. Question/Discussion What can we say about the surface area of cubes and rectangular prisms? Answer Surface area is about covering or wrapping something. When disassembled, rectangular prisms are made up of connected views, or faces. The faces can be assembled in a net (orthographic map) to show the 3 dimensional shape on a 2-dimensional plane. The net can be reassembled into a 3-dimensional shape. We can determine surface area by summing all the areas of all the faces. Surface area is labeled in square units. Question How would surface area be important to NASA scientists and engineers working on the PSA? Answer NASA would need to know how much space they needed to cover or package, how large or small PSA was going to be, how much they could spend on supplies and materials, etc. Elicit student responses and encourage discussion. Question What relationships did we discover existed between cubes and rectangular prisms?

Answer Cubes - all faces are the same size. Rectangular prisms - all faces are different sizes. Nets can be used to understand surface area for cubes and rectangular prisms. Question/Discussion What can we say about the volume of cubes and rectangular prisms? Answer Volume is about holding something in a space. Volume is about 3-dimensional measurements. Using the number of cubes that fit into each layer, row, and column inside the cereal box we were able to determine the volume of the box. The number of cubes in each layer tells us the height of the box. The number of cubes in each row tells us the width of the box. The number of cubes in each column tells us the length of the box. Multiplying the height times the width times and length tells us the volume of the box. Question How would volume be important to NASA scientists and engineers working on the PSA? Answer NASA would need to know how much space was available for PSA components and materials, etc. Elicit student responses and encourage discussion. Explain We know that in math we can determine important ideas by using formulas (an equation that states a rule or fact). For example we know that in order to determine area of a quadrilateral we can say: Area = length x width Question How else can we usefully use this formula?

Answer To determine surface area of a cube or rectangular prism - when all areas are known they can be summed to determine the total surface area. Explain/Question Just like the formula A=length x width tells us the area in square units for quadrilaterals what else can we can use formulas for? Answer To determine other shape measurements. Explore Explain In the previous investigations we used nets to show surface area by enclosing and measuring a space. We used boxes and cubes to show

volume by filling and measuring a space. Question What is the surface area of a cube? Answer The surface area of a cube is the area of the six squares that cover it. Explain The area of one of the squares is (a*a) or (a2). (“a” is the length of the side of each edge of the cube).

a a

Explain Since these areas are all the same, you can multiply one of them by six. So the surface area of a cube is 6 times one of the sides squared. Surface Area of a Cube = 6 a 2 We say squared since the exponent in the formula indicates repeated multiplication. (a times a, or a times itself) Explain We use the number 2 for the exponent in the formula since we are multiplying two dimensions thus finding an area. Area is expressed in units squared since it is the number of square units that cover a given shape or figure. Rectangular S. Area = 2ab + 2bc + 2ac (a, b, c are the lengths of the 3 sides) So, the surface area of a rectangular prism is the area of the six rectangles that cover it. But we don't have to figure out all six because we know that the top and bottom are the same, the front and back are the same, and the left and right sides are the same.

The area of the top and bottom (side lengths a and c) = a*c. Since there are two of them, you get 2ac.

The front and back have side lengths of b and c. The area of one of them is b*c, and there are two of them, so the surface area of those two is 2bc.

Explain The left and right side have side lengths of a and b, so the surface area of one of them is a*b. There are two of them, so their combined surface area is 2ab. Rectangular Surface Area = 2ab + 2bc + 2ac

Question What is the volume of a rectangular prism? Answer The volume of a rectangular prism is the measurement of available space inside the prism. Explain Volume of a rectangular prism = a*b*c units cubed. (length ‘a’ x width ‘b’ x height ‘c’) Area is measured in "cubic" units. The volume of a figure is the number of cubes required to fill it completely, like cubes in a box. We say cubed since the exponent in the indicated repeated multiplication. (length x width x height) We use the number 3 for the exponent in the formula since we are multiplying three dimensions thus finding a volume. Volume is expressed in units cubed since it is the number of cubic units that fill a given shape or space (like cubes in a box). Teacher Action To each student team member distribute small rectangular boxes,

scissors, drawing paper and metric rulers to students. Instruction Students will remove the tape or other material holding the rectangular box together and spread out the parts onto the drawing paper on a flat surface and sketch or trace the net of the rectangular box. Student Action Fold out or cut out the box into a flat shape and sketch or trace the net of

the box. Instruction Have students measure the length of each side of each face of the open

box and record results in Math Journals.

Student Action Students measure and record each face of the open box. Instruction Have students measure one of the ends of the box and record results in

Math Journals. Student Action Students measure and record one of the ends of the open box. Instruction Have students measure the front of the box and record results in Math Journals. Student Action Students measure the front of the box and record results in Math Journals.

Back view__________cm. Left side view_______cm. Top view___________cm. Front view__________cm. Bottom view________ cm. Right side view_______cm. Instruction Have students label the measurements on the box with a pen or marker. Question What have determined so far with our measurements? Elicit student response & discussion. Answer The 2-dimensional measurements of each face of the box. Question Are some of the measurements the same? If so, which ones? Elicit student response & encourage discussion.

Answer Just as in the previous investigations the properties of a cube determine that it will have identical measurements on all its sides. Additionally, the properties of a rectangular prism determine that opposite faces will have identical measurements on opposite sides.

Instruction/Question Within your team talk about and be prepared to report on and justify the following questions. Are there faces of the box that have exactly the same measurements? Does this apply to any rectangular prism? Note to Teacher Allow time for student teams and whole group to adequately make

and justify their positions. Answer Yes, there are faces of the box that have exactly the same measurements – this does not apply to any rectangular prism. For example, a single serving cereal box has the same measurements on opposite faces, but not the same measurements on all faces. A cube shaped box has the same measurements on all faces. Encourage discussion & connections to previous investigations.

Explain Work with your group to find the area of each section of the box, add them together and find the total surface area of the box. Back view__________cm. Left side view_______cm. Top view___________cm. Front view__________cm. Bottom view________ cm. Right side view_______cm

Total surface area_________sq. cm.

Teacher Action Have students report the surface area they found and turn the box over for re-use with a second exploration. Explain We’re going to re-use our rectangular prisms to investigate volume. Question What are some possibilities about why the idea of re-use or recycling is important to NASA astronauts on board the ISS, and to scientists

supporting them from Earth? Elicit student response and facilitate discussion.

Answer On the ISS there’s a limited amount of space, supplies, and materials that have to last for long periods of time. Re-using or recycling allows supplies and materials to extend there use-time, which saves money, and time,

etc…

Teacher Action Have students reassemble their rectangular prisms with tape. Instruction In your math journal give an estimate of the volume of the small box. Use your previously collected surface area measurement data to determine the volume of the box. Record these results in your math journal.

Question What was the volume of your box? Teacher Action

Elicit student response and discussion. Explain Question What does the number we found for total surface area tell us? Answer The number we found tells us total surface area of the box – the total measurement of 2-dimensional space that can cover the of the surface. Question What does the number for we found volume tell us?

Answer The number we found tells us the volume of the box - the measurement of available space inside the box. Question Why do we use square units or units squared to express surface area measurements? Answer It shows the measurement of two-dimensional, or flat surface dimensions

of length and width. It shows how many 1 x 1 unit squares are needed to

cover or wrap an object. Questions Why do you use cubic units or units cubed to express volume measurements? Answer Volume is expressed in units cubed since it is the number of cubic units

that fill a given shape or space (like how many cubes fit in a box). Answer It shows how many 1 x 1 x 1 unit cubes are needed to fill an object. Question What types of relationships exist between surface area and volume? Answer Since they use different measuring units it can be challenging to compare surface area and volume. (It's like asking why I am sometimes older than my weight). Volume looks at how much space, or capacity is inside of a 3-dimensional object. Surface area looks at the amount of enveloping 2-dimensional surface space on the outside of the 3-dimesnisonal object.

Question How do surface area and volume affect one another? Answer Surface area and volume change disproportionately – as one changes in size the other changes in size by a different factor. Surface area and volume change by different amounts.

When Volume is constant, the maximum surface area occurs with the most elongated package and minimum surface area occurs with the most cube-like

Constructed Response Performance Task Explain/Instruct Discuss with your team the following three topics: 1. How NASA scientists working with the PSA Project might make effective use of surface area and volume when designing and building the PSA. 2. Discuss with your team when in the real world you might apply the concepts of surface area and volume. Explain why you’d chose to find volume and surface area as opposed to just finding area. 3. Imagine you are telling someone else how to figure surface area and volume what do you tell them? Practice this scenario with your team. Explain Use your math journal to record all data to demonstrate your understanding of views and nets by creating an example of your own (actually construct the 3-D figure with graph paper, scissors, and tape) Explain/Instruct Sketch the 3-D view of the rectangular prism OR the net of it, measure it’s dimensions and label them on your sketch. In an organized manner, show all the steps to calculating the surface area and volume of the figure. Student Action Collaborate within teams to logically discuss given topics. Complete the 3-D figure construction and measurement project.

Teacher Action Facilitate student team discussions. Maintain observational records

journal. Evaluate Discussion Student team conversations/insights from topics (1, 2, and 3). Check for student understanding. Teacher Action Apply observational records journal anecdotes from student discussions

on topics (1, 2, and 3) as assessment. Collect and review student

constructed figures and math journals for use as assessment. Evaluate for accurate measurements and concept applications.

Evaluative Rubric Criteria Guide

Scoring Levels Level 6

Level 5 Level 4 Level 3 Level 2 Level 1

Mastery of Topics

Mastery of all topics

Mastery of almost all topics

Mastery of most of the topics

Mastery of several of the topics

Mastery of few of the topics

Limited understanding of only a few of the topics

Solutions Explanation

Clear, convincing explanation of all solutions

Clear, convincing explanations of most solutions

Clear, convincing explanations of many solutions

Clear, convincing explanations of several solutions

Clear, convincing explanations of a few solutions

Partial explanations of a few solutions

Errors Committed

Almost no errors

Few errors Some errors, most of which are minor

Significant number of errors, some of which are major

Many, errors, including several major ones

Errors throughout the entire assessment, including many major ones

Steps Shown

All steps shown in each solution

All steps shown in most solutions

All steps shown in many solutions

All steps shown in several solutions; most steps shown in other solutions

All steps shown in a few solutions; some steps shown in other solutions

Some steps shown in some solutions

Response to Open Questions

Excellent, accurate responses to open-ended questions

Satisfactory and accurate responses to open-ended questions

Satisfactory responses to some of the open-ended questions

Satisfactory response to at least one open-ended question

An attempt to respond to at least one open ended question

Inappropriate responses, or no responses, to all open-ended questions

Response to Problems

Correct, complete responses to almost all of the more difficult problems and all basic problems

Correct, complete solution to many of the more difficult problems and all basic problems

Correct, complete solutions to several of the more difficult problems and all basic problems

Attempts solutions to some of the more difficult problems and correctly completes basic problems

Completes most basic problems correctly

Has difficulty completing any basic problems correctly

References and Resources NASA Intelligent Systems Project NASA Computing, Information, and Technology Program NASA Engineering Complex Systems (ECS)

NASA Cross Enterprise Technology Development Program Gregory A. Dorias and Yuri Gawdiak, “The Personal Satellite Assistant: An Internal Spacecraft Autonomous Mobile Monitor.” White paper for IEEEAC. NASA Ames Research Center, (2003). “NASA Ames Research Center Personal Satellite Assistant” Project Website. http://quest.nasa.gov/projects/psa/, (2003). “NASA Building a Droid for the ISS” Project Website. http://science.nasa.gov/headlines/y2001/ast23jul_1.htm “NASA Temporary Weightlessness” Website http://science.nasa.gov/newhome/headlines/fluid/zero-g.plane.htm “NASA What’s Shaking in Space?” http://science.nasa.gov/newhome/headlines/msad15jul97_1.htm “NASA Micro gravity Takes a Quantum Leap” http://science.nasa.gov/newhome/headlines/msad15jul98_1.htm “NASA International Space Station” http://science.nasa.gov/headlines/y2000/ast24jul_1m.htm http://science.nasa.gov/headlines/y2000/ast02aug_1.htm “NASA Breathing Easy on the Space Station” http://science.nasa.gov/headlines/y2000/ast13nov_1.htm PSA Semi-Annual Internal Review & Interview NASA Ames Research Center, (2003).